Sparse Dual Frames and Dual Gabor Functions of Minimal Time and Frequency Supports

Abstract

Exploitation of the optimality of (non-exact) frames from a sparse dual point of view is presented. Sparse dual frames and dual Gabor functions of the minimal time and/or frequency supports are studied and constructed through the notion of sparse representations. Conditions on the sparsest dual frames and the dual Gabor functions of the minimal time and/or frequency supports are discussed. Algorithms and examples are provided.

Appendices

Appendix A: Proof of Theorem 3.1

Proof

Since {g _m,n } and {γ _m,n } are a pair of Gabor frame and dual Gabor frame for ℂ^d,

$$ \forall f \in\mathbb{C}^d,\quad f = \sum_{n=0}^{N-1} \sum _{m=0}^{M-1} \langle{f}, {\gamma_{m,n }} \rangle g_{m,n}. $$

(A.1)

We can rewrite (A.1) as, for all t=0,…,d−1,

which is equivalent to, for all t,τ=0,…,d−1,

$$\sum_{n=0}^{N-1} \sum _{m=0}^{M-1} \gamma^*_{m,n}(\tau) g_{m,n}(t) = \delta(t-\tau). $$

Substituting expressions of γ _m,n (⋅) and g _m,n (⋅) into the left hand side, we have, for t,τ=0,…,d−1,

(A.2)

Based on the orthogonality relation

$$ \sum_{m=0}^{M-1} e^{ 2\pi i (t-\tau) \frac{m }{M} } = \left \{ \begin{array}{l@{\quad}l} M, & \hbox{$(t-\tau) \bmod M= 0$;} \\ 0, & \hbox{otherwise.} \end{array} \right . $$

(A.3)

The previous system can be reduced further.

Fix τ∈{0,…,d−1}, for t=0,…,d−1, when (t−τ)modM≠0, (A.2) (A.2) holds automatically. When t=τ, (A.2) is reduced to

$$\sum_{n=0}^{N-1} \gamma^{*}( \tau-na) g(t - na) = \frac{1}{M}. $$

When (t−τ)modM=0 and t≠τ, (A.2) is reduced to

$$\sum_{n=0}^{N-1} \gamma^{*}( \tau-na) g(t - na) = 0. $$

A combination of all scenarios above leads to the proposed duality condition. □

Appendix B: Proof of Lemma 4.1

Proof

Given any $h \in\operatorname{ker}(A)\backslash\{ 0 \}$, based on the support of x, ∥x+h∥₁ can be divided into two parts, i.e.,

Since $\sum_{t \in T} h(t) \operatorname{sgn}( x )(t) + \sum_{t \in T^{C}} |h(t)| > 0$, we see that

$$\| x + h \|_{1} \geq\| x \|_{1} + \sum _{t \in T} h(t) \operatorname{sgn}( x ) (t) + \sum _{t \in T^C} \bigl\vert h(t)\bigr\vert> \| x \|_{1}. $$

Namely, x is the unique solution to (P ₁).

For the other direction, assume there exists some $h_{0} \in \operatorname{ker}(A) \backslash\{ 0 \}$ satisfying

$$\sum_{t\in T} h_0(t) \operatorname{sgn}( x ) (t) + \sum_{t\in T^C} \bigl\vert h_0(t)\bigr\vert\leq0. $$

We choose a sufficiently small ϵ>0 such that, for all t∈T,

$$\bigl\vert x(t)\bigr\vert+ \epsilon h_0(t) \operatorname{sgn}( x ) (t) \geq0. $$

Then

Since $\sum_{t\in T} h_{0}(t) \operatorname{sgn}( x )(t) + \sum_{t\in T^{C}} |h_{0}(t)| \leq0$, and ϵ>0,

$$\| x + \epsilon h_0 \|_{1} \leq\| x \|_{1}, $$

which is contradictory to x being the unique solution to (P ₁). The claim then follows. □